I posted a note on this a few days ago on the SD state trip reports section of this forum. I had been staring at a relief map (apparently a little too long) and projecting the longitude line south from Harney. It sure looked likely that Texas had mountainous areas as high and as far east as the Black Hills.
Roger's America's Roof website has lists of the highest peaks in each state with lat/long info.
There are five Texas (Big Bend) peaks that are both higher and farther east than Harney. Some judge these peaks to be part of the Rockies. If so, Harney would still lose claim to the highest U.S. point east of the Rockies, as it isn't really east of ALL of the Rockies.
It appears as though the farthest point east that is still higher than Mt. Mitchell, NC is Crown Peak, TX (about 7,000' and roughly 10 miles east of Harney Peak).